TORIC STRUCTURE ON MUMFORD-TATE DOMAINS AND CHARACTERISTIC COHOMOLOGY 5 1 0 MOHAMMAD REZA RAHMATI 2 b Abstract. WeexplainahigherstructureonKato-UsuicompactificationofMumford- e Tate domains as toric stacks. As a motivation the universal characteristic coho- F mology of Hodge domains can be described as cohomology of stacks which have 2 better behaviour in general. 1 ] G A . h 1. Introduction t a m Assume D is a period domain of pure Hodge structures defined by Griffiths, with [ the period map (∆∗)n → Γ \ D, where Γ is the monodromy group. We treat a 6 partial compactification of Γ\D so that the period map is extended over ∆n. Kato v 6 and Usui generalize the toroidal compactifications for any period domain D, which 8 is not Hermitian symmetric in general and show these are moduli spaces of log 8 Hodgestructures. This partial compactification is given by using toroidal embedding 6 0 associated to the cone generated by the data of the monodromies, [TH]. . 1 0 Mumford-Tate domain can be considered as a toric stack, by a definition using 5 1 Stacky-fans. This structure is compatible with the Kato-Usui generalized toroidal : compactification. In this way we may regard some additional data on MT-domains v i which preserves the stabilizers of the points under a Lie group action. There are X several definitions of toric Stacks. A definition due to Lafforgue, defines a toric stack r a to be the stack quotient of a Toric variety by its torus. The second, Borisov-Chen- Smith, define smooth toric Deligne-Mumford stacks. These stacks have smooth toric varieties as their coarse moduli space. The third, due to Tyomkin, includes all toric varieties, which can be singular, [GS]. We define a toric stack to be the stack quotient of a normal toric variety X, by a subgroup G of its Torus T , cf. [GS]. We begin with the ordinary definitions in the 0 theory of toric varieties, and then try to upgrade the definition for higher structures. The main task of this text is to explain how to express the definition of a toric Keywordsandphrases. Mumford-Tatedomains,Boundarycomponentsorperioddomains,Toric Stacks, Stacky-fans. 1 2 MOHAMMAD REZA RAHMATI stack for period domains of Griffiths, Mumford-Tate domains, their boundary points via the Kato-Usui generalized toroidal compactification. As an application one can consider the universal characteristic cohomology of Mumford-Tate domains as a type of stack cohomology. 2. Toric varieties and the moment map In this section we briefly explains basic definitions in the theory of toric varieties mainly extracted from [WF]. Toric varieties primarily came up in connection with the study of compactification problems. This compactification simply says a toric varietyX, isanormalvarietythatcontainsatorusT asadenseopensubset, together with an action (1) T ×X −→ X of T on X that extends the natural action of T on itself. The torus T is C∗×...×C∗. The simplest example is the projective space Pn as the compactification of Pn. There are several equivalent ways to construct such varieties, that we explain some of them below. A toric variety may be constructed from a lattice N which is a collection of strongly convex rational polyhedral cones, σ in the real vector space N = N ⊗ R. Let R M = Hom(N,Z) denote the dual lattice with dual pairing h.,.i. If σ is a cone in N, let σ∨ be its dual in M . Then setting, R (2) S = σ∨ ∩M =: {u ∈ M | hu,vi ≥ 0 for all v ∈ σ} σ defines the toric variety as (3) U = Spec(C[S ]) σ σ where by C[S ] we mean C[x[Sσ]]. If τ is a face of σ then the above procedure σ identifies U ֒→ U as an open subset. Thus the faces of σ provide an open cover τ σ of U . A basic example is to take the cone to be 0 ∈ Rn. It is a face of every σ other cone. The dual lattice M is generated by the standard generators ∓e ,...,∓e . 1 n Take X ,...,X the elements corresponded to the dual basis in C[M]. The C[M] = 1 n C[X ,X−1,...,X ,X−1]whichistheaffineringofthetorusU = T = (C∗)n. Trivially 1 1 n n 0 every toric variety contains the torus as an open subset. TORIC STRUCTURE ON MUMFORD-TATE DOMAINS AND CHARACTERISTIC COHOMOLOGY3 For any cone σ in a lattice N, the corresponding affine variety U has a distinguished σ point which we denote by x . This point in U is given by a map of semi-groups σ σ 1 if u ∈ σ⊥ (4) u → 0 otherwise ( The point x is independent of γ ⊃ σ. σ GivenalatticeN withdualM, wehavethecorrespondingtorusT = Hom(M,G ). N m ThenoneknowsthatHom(G ,T ) = Hom(Z,N) = N. ThetorusT = Spec(C[M]) m N N = Hom(M,C∗) = N ⊗C∗ acts on U as follows. A point t ∈ T is identified with a σ N map M → C∗, and a point x ∈ U with a map S → C of semigroups. Then t.x is σ σ the map of semigroups u → t(u)x(u). It follows that, every 1-parameter subgroup λ : G → T isgiven byaunique v ∈ N. Wedenotebyλ the1-parametersubgroup m N v corresponding to v. DuallyHom(T ,G ) = Hom(N,Z) = M. Every character χ : T → G is given by N m N m auniqueu ∈ M. Thecharactercorrespondingtoucanbeidentifiedwiththefunction zu = χu in the coordinate ring C[M] = Γ(T ,O∗). Then λ (z)(u) = χu(λ (z)) = N v v zhu,vi. As with any set on which a group acts, a toric variety X is the disjoint union of its orbits by the action of the torus T . There is one such orbit O for each cone τ ∈ ∆. N τ If τ has maximal rank, then O is a point x . If τ = 0 then O = T . O is an open τ τ τ N τ subvariety of its closure V(τ), which is a closed toric subvariety of X. V(τ) is the disjoint union of those orbits O for which γ ⊃ τ. Therefore γ (5) O = T = Hom(M(τ),C∗) = Spec(C[M(τ)]) = N(τ)⊗C∗ τ N(τ) This is a torus of dimension n−dim(τ), on which T acts by T → T . N N N(τ) The star of τ is defined as the set of cones σ in ∆ that contain τ as a face. Such cones σ are determined by their images in N(τ), (6) σ¯ = σ +(N ) /(N ) ⊂ N /(N ) = (N ) τ R τ R R τ R τ R These cones{σ¯ : τ ⊂ σ}formafaninN(τ), andwedenotethisfanbyStar(τ). One knows that V(τ) = X(Star(τ)). A cone σ is called non-singular if it is generated by part of a basis for the lattice N. This implies the affine toric variety is non-singular. 4 MOHAMMAD REZA RAHMATI 3. Stacks vs Artin and Deligne-Mumford Stacks In this section we discuss what is the point of view in compairing the ordinary notion of schemes with the corresponding stack object. We avoid of having a strict mathematical language, and try to give the intuitive idea behind the concept of a stack. This section is a brief from the article [BN] which I suggest to every one to look at. Stacks were introduced by Grothendieck to provide a general framework for study- ing local-global phenomena in mathematics. The early stages of the development of the theory can be traced out in the Ph.D thesis of the students of Grothendieck’s students. Later it was used by Deligne and Mumford, that what is now called Deligne-Mumford stack. Later E. Artin generalized Deligne-Mumford work in which it became a vital tool in algebraic geometry, specially in the study of quotient spaces. Every scheme is a Deligne-Mumford (DM-)stack and every DM-stack is an Artin stack, in which all are also called algebraic stacks. Algebraic stacks are a new breed of spaces for algebraic geometer, providing greater flexibility for performing con- structions which are impossible in the category of schemes. The notions of analytic, differentiable, and topological stacks was introduced analogously in the correspond- ing categories. A toplogical space is naturally a topological stack. Main classes of examples can be obtained from a topological group acting continuously on a topological space X, in which we associate what is called the quotient stack of the action, and is denoted by [X/G]. The quotient stack [X/G] is better behaved than X/G, and retains much more information, specially when the action as fixed points or misbehaved orbits. For instance [X/G], in some sense remembers all the stabilizer groups of the action, while X/G is blind to them. There is a natural morphism π : [X/G] → X/G mod which allows us to compare the stack [X/G] with the coarse moduli X/G. One can produce lots of examples by gluing quotient stacks. We call such stack locally quotient stack. Such topological stacks are called Deligne-Mumford. A topological stack is uniformizable if it is of the form [X/G], where G is a discrete group acting properly discontinuously on a topological space X. Thus every DM-stack is locally uniformizable. There are examples of DM-stacks that are not globally uniformizable. Every toplogical stack X has an underlying topological space called coarse moduli space denoted X . There is a natural functorial map π : X → X called the mod mod mod moduli map. Roughly X is the best approximation of X by a topological space. mod Assume that X = [X/G] is a quotient stack. Then there is a natural quotient map q : X → [X/G], and this maps makes X a principal G-bundle over [X/G]. So in particular q : X → [X/G] is a Serre fibration. the usual quotient map we know TORIC STRUCTURE ON MUMFORD-TATE DOMAINS AND CHARACTERISTIC COHOMOLOGY5 from topology is the composition π ◦ q. A basic example is when a topological mod group G acts on a point ∗, trivially. The quotient stack [∗/G] of this action is called classifying stack of G, and is denoted BG. Note that (BG) is a point. However mod BG is far from being a trivial object. More precisely, the map ∗ → BG makes ∗ into a principal G-bundle over BG, and this universal G-bundle is universal. That is for every topological space T, the equivalence classes of morphisms T → BG are in bijection with the isomorphism classes of principal G-bundles over T. In this situation if G is discrete, the quotient map ∗ → BG becomes the universal cover of BG. We can think of X as a topological space X which at every point x is decorated mod with a topological group I . The group I is called the stabilizer or inertia group x x at x. These inertia groups are interwined in an intericate way along X . When mod X is Deligne-Mumford, all I are discrete. At every point x ∈ X we have a pointed x map (BI ,x) → (X,x). A basic example can be the shpere S2 with an action of Z x n as rotations fixing the north and south poles. the stack [S2/Z ] has an underlying n space which is homeomorphic to a sphere, however [S2/Z ] remembers the stabilizers n at the two fixed points, namely the north and the south poles. Thus [S2/Z ] is like n BZ at the fixed points and in the remaining points is like the sphere. n Agroupoidisacategorysuchthatanymorphismbetween twoobjects isinvertible. If a group acts on a set X one may form the action groupoid representing this groupoid action, by taking the objects to be the elements of X and the morphisms to be elements g ∈ G such that g.x = y and compositions to come from the binary operation in G. More explicitly the action groupoid is the set G×X (often denoted G ⋉ X) such that the source and target maps are s(g,x) = x, t(g,x) = gx. The action ρ is equivalently thought of as a functor ρ : BG → Sets, from the group G regarded as a one-object groupoid, denoted by BG. This functors sends the single object to the set X. let Set be the category of pointed sets and Set → Sets be the ∗ ∗ forgetful functor. We can think of this as a universal set-bundle. Then the action groupoid is the pullbak [X/G] −−−→ Sets ∗ (7)   BG −−−→ Sets   y y Thenotionofgroupoidsandtheiractioncanbegeneralizedoverarbitraryschemes, as Picard stacks with similar definitions, [FMN]. 6 MOHAMMAD REZA RAHMATI 4. Toric Stacks by Stacky-fans Toric stacks are examples of locally quotient stacks having toric structures. There are several ways to express the definition, however we follow the reference [GS]. A toric stack is the stack quotient of a normal toric variety X, by a subgroup G of its torus T. The stack [X/G] has a dense open torus T = T/G which acts on [X/G]. Such toric stacks have trivial generic stabilizers. Definition 4.1. A toric stack is an Artin Stack of the form [X/G], together with the action of the torus T = T/G. A non-strict toric stack is an Artin stack which is isomorphic to an integral closed torus-invariant substack of a toric stack, i.e. is of the form [Z/G], together with the action of the stacky torus [T/G]. Taking G to be trivial gives rises to the definition of a Toric variety. Also taking G = T gives the Lafforgue’s definition stated in the introduction. Definition 4.2. A stacky fan is a pair (Σ,β), where Σ is a fan on a lattice L and β : L → N is a homomorphism to a lattice N, so that coker(β) is finite. A stacky fan gives rise to a toric stack as follows. Let X be the toric variety Σ associated to Σ. The map β∗ : N∗ → L∗ induces a homomorphism of Tori, T : β T → T , by naturally identifying β with the induced map on lattices of 1-parameter L N subgroups. Since coker(β) is finite, β∗ is injective, so T is surjective. Let G = β β kerT . Note that T is the torus on X , and G ⊂ T is a subgroup. The action of β L Σ β L G on X is induced by the homomorphism G → T . β Σ β L Definition 4.3. If (Σ,β) is a stacky fan, we define the toric stack X to be [X/G ], Σ,β β with the torus T = T /G . N L β Every toricstack arisesfromastacky fan, sinceevery toricstackisoftheform[X/G], where X is a toric variety and G ⊂ T is a subgroup of its torus. Associated to X is 0 a fan Σ on the lattice L = Hom(G ,T ). The surjection of tori T → T/G induces a m 0 homomorphism of lattices of 1-parameter subgroups, β : L → N := Hom(G ,T/G). m The dual homomorphism β∗ : N∗ → L∗ is the induced homomorphism of characters. Since T → T/G is surjective, β∗ is injective, and the cokernel of β is finite. Thus (Σ,β) is a stacky fan and [X/G] = X . Σ,β Example 4.4. Take Σ to be an arbitrary fan on a lattice N, L = N, and β = id. The induced map T → T is also identity and G = 0. Then X is the toric N N β Σ,β variety X . Σ TORIC STRUCTURE ON MUMFORD-TATE DOMAINS AND CHARACTERISTIC COHOMOLOGY7 (1 1) Begin with X = C2 \(0,0) and β : Z2 −→ Z. The induced map by β∗ : Z → Z2 Σ with G2 → G is given by (s,t) 7→ st−1. Thus G = G = {(t,t)} ⊂ G2 . Then m m β m m X = P1. Σ,β A T-invariant substack of [X/G] is necessarily of the form [Z/G], where Z ⊂ X is an integral T-invariant subvariety of X. The subvariety Z is naturally a toric variety whose torus T′ is a quotient of T. The quotient stack [Z/G] contains a dense open stacky torus [T′/G] which acts on [Z/G]. A morphism of toric stacks is a morphism which restricts to a homomorphism of stacky tori and is equivariant with respect to that homomorphism. A morphism of stacky fans (Σ,β : L → N) → (Σ,β : L′ → N′) is a pair of group homomorphisms Φ : L → L′ and φ : N → N′ so that β′◦Φ = φ◦β and so that for every σ ∈ Σ, Φ(σ) is contained in a cone of Σ′. We draw this morphism as Σ → Σ′ L −−Φ−→ L′ β β′ N −−−φ→ N′ y y A morphism of stacky fans (Φ,φ) : (Σ,β) → (Σ′,β′) induces a morphism of toric varieties XΣ → XΣ′ and a compatible morphism of groups Gβ → G′β, so it induces a toric morphism of toric stacks X(Φ,φ) : XΣ,β → XΣ′,β′. Definition 4.5. (Fantastacks) Let Σ be a fan on a lattice N, and let β : Zn → N a homomorphism with finite cokernel so that every ray of Σ contains some β(e ), and i every β(e ) lies in the support of Σ. For a cone σ ∈ Σ, let σˆ = cone({e | β(e ) ∈ σ}). i i i We define the fan Σˆ on Zn to the fan generated by all σˆ. Define F = X . Any Σ,β Σ,β stack isomorphic to F is called a Fantastack. Σ,β A simple example is to take Σ the trivial fan on N = 0, and β : Zn → N to be the zero map. Then Σˆ is the fan of Cn, and G = Gn. So F = [Cn/Gn]. A sort β m Σ,β m of examples are any smooth toric variety X where β : Zn → N is constructed by Σ sendingthegeneratorsofZn tothefirstlatticepointsalongtheraysofΣ. ThenX = Σ F . The cones of Σˆ are indexed by sets {e ,...,e } such that {β(e ),...,β(e )} Σ,β i1 ik i1 ik is contained a simple cone of Σ. It is then easy to identify which open subvariety of Cn is represented by Σˆ. Explicitly define the ideal 8 MOHAMMAD REZA RAHMATI I = ( x | σ ∈ Σ) Σ i β(Yei)∈/σ Then X = Cn \V(I ). Σˆ Σ There exists an alternative definition for toric stacks generalizing the torus action to higher structures. In the following definition set T := Hom(A,C∗) for an abelian A group A( the group of characters ). A Deligne-Mumford torus is a Picard stack over Spec(C) which is obtained as a quotient [T /G ], with φ : L → N is a morphism of finitely generated abelian groups L N such that ker(φ) is free and coker(φ) is finite. Any Deligne-Mumford (DM) torus is isomorphic as Picard stack to T ×BG, where T is a torus and G is a finite abelian group. Then, asmoothtoricDeligne-MumfordstackisasmoothseparatedDM-stack X together with an open immerssion of a Deligne-Mumford torus ı : T ֒→ X with dense image such that the action of T on itself extends to an action T ×X → X. In this case a morphism is a morphism of stacks which extends a morphism of Deligne- Mumford tori. A toric orbifold is a toric DM-stack with generically trivial stabilizer. A toric DM-stack is a toric orbifold iff its DM-torus is an ordinary torus, [FMN]. 5. Cohomology of Deligne-Mumford (DM)-Stacks In this section we provide the definition of the de Rham cohomology of stacks for an application to characteristic cohomology of Mumford-Tate domains. This brief is taken from the lectures of K. Behrend at UBC [KB]. Differentiable stacks are stacks over the category of differentiable manifolds. They are the stacks associated to Lie groupoids. A groupoid X ⇒ X is a Lie groupoid if X and X are differentiable 1 0 0 1 manifolds, structure maps are differentiable, source and target maps are submersion. There is associated a simplicial nerve to a Lie groupoid namely p times X := X × X × ...× X p 1 X0 1 X0 X0 1 z }| { Then we get an associated co-simplicial object p Ωq(X ) → Ωq(X ) → Ωq(X ) → ... , ∂ = (−1)i∂∗ 0 1 2 i i=0 X TORIC STRUCTURE ON MUMFORD-TATE DOMAINS AND CHARACTERISTIC COHOMOLOGY9 the cohomology groups Hk(X,Ωq) are called the Cech cohomology groups of the groupoid X = [X ⇒ X ] with values in the appropriate sheaf. 1 0 Definition 5.1. Let X be a differentiable stack. Then Hk(X,Ωq) = Hk(X ⇒ X ,Ωq) 1 0 for any Lie groupoid X ⇒ X giving an atlas for X. In particular this defines 0 Γ(X,Ωq) = H0(X,Ωq) Example 5.2. If G is a Lie group then Hk(BG,Ω0), is the group cohomology of G calculated with differentiable cochains. Definition 5.3. The double complex Apq := Ωq(X ) is called the de Rham complex p of the stack X, and its cohomologies are called de Rham cohomologies of X. One can show using a double fibration argument to prove that the de Rham coho- mologyisinvariant under Moritaequivalence andhence well defined fordifferentiable stacks. Thus Hn (X) = Hn (X ⇒ X ), for any groupoid atlas X ⇒ X of the DR DR 1 0 1 0 stack X. When the stack X is a quotient stack there is a clear explanation of its cohomology as an equivariant cohomology. There is a well-known generalization of the de Rham complex to the equivariant case, namely the Cartan complex Ω•(X), defined by G (8) Ω•(X) := (Skg∨ ⊗Ωi(X))G G 2k+i=n M where S•g∨ is the symmetric algebra on the dual of the Lie algebra of G. The group G acts on g by adjoint representation on g∨ and by pull back of differential forms on Ω•(X). The Cartan differential is d − ı where ı is the tensor induced by the DR vector bundle homomorphism g → T coming from differentiating the action. If G X X is compact, the augmentation is a quasi-isomorphism, i.e (9) Hi (X) −∼=→ Hi(Tot Ω•(X )) G G • 10 MOHAMMAD REZA RAHMATI for all i and the groupoid X . • Proposition 5.4. [KB] If the Lie group G is compact, there is a natural isomorphism Hi (X) −∼=→ Hi (G×X ⇒ X) = Hi ([X/G]) G DR DR As a corollary H∗ (BG) = (S2∗g∨)G for a compact lie group G. DR Remark 5.5. [KB] If G is not compact, then H ([X/G]) is still equal to equivariant DR cohomology. This fact holds for equivariant cohomology in general. As in the de Rham cohomology, every topological groupoid defines a simplicial nerve X which gives rise to the double complex C (X ) of simplices. The total homology • q p of this double complex are called singular homology of X ⇒ X . A simple example 1 0 is to consider the transformation groupoid G×X ⇒ X for a discrete group G. In this case we have the degenerate spectral sequence (10) E2 = H (G,H (X)) ⇒ H (G×X → X) p,q q p p+q When X is a point H (G × X ⇒ X) = H (G,Z). In general there exists inter- p p pretation of the singular homology H ([X/G]) in terms of the equivariant homology ∗ of X when the Lie group G acts continuously on X, exactly similar to de Rham cohomology case. There is also the dual notion of singular cohomologies of the stack X by replacing the double complex C (X ) with its dual Hom(C (X ),Z). Directly q p q p we obtain a pairing (11) H (X,Z)×Hk(X,Z) → Z k Example 5.6. [KB] The stack of triangles up to similarity may be represented by S ×∆ ⇒ ∆ . Thus the homology of the stack of triangles is equal to the homology 3 2 2 of the symmetric group S . 3 Similarly the stack of Elliptic curves M may be represented by the action of Sl (Z), 1,1 2 by the linear fractional transformations on the upper half plane in C. Thus the homology of the stack of Elliptic curves is equal to the homology of Sl (Z). 2